I'm thinking about these statements and I would like to know if I am right.
A principal ideal ring $R$, by definition, is a ring whose ideals are principals, since $R$ is itself an ideal, $R=(x)$, for some $x\in R$.
If $R=(y)$ for some $y$, every ideal of $R$ is principal.
Using (1) and (2), the principal ideal rings are in this form $(x)$, for some element $x$ and vice and versa, every (x) is a principal ideal ring.
Am I wrong? I need some help.
Thanks a lot